Possible method to prove infinite twin prime conjecture

Wolfram Language code to test whether does $p$ provide the twin prime number pair via your conjecture or not is as follows.

twinPrimesQ[tp_]:=tp[[1]]+2==tp[[2]]&&PrimeQ[tp[[1]]]&&PrimeQ[tp[[2]]];
primesList[p_]:=Module[{out={Prime[3]},i},
    For[i=4,Prime[i]<=p,i=i+1,
        out=Append[out,Prime[i]];
    ];
    out
];
testPrime[p_,pl_]:=Module[{out,found=False,twinPrimes,primeFactors,primeFactorsPowers,i},
    twinPrimes={
        {3 5 prod p-4,3 5 prod p-2},
        {3 5 prod p+2,3 5 prod p+4}
    };
    primeFactors=primesList[p];
    primeFactorsPowers=Tuples[Range[0,pl],primeFactors//Length];
    For[i=1,i<=Length[primeFactorsPowers],i=i+1,
        out=twinPrimes/.prod->Product[primeFactors[[k]]^primeFactorsPowers[[i]][[k]],{k,1,primeFactors//Length}];
        found=twinPrimesQ[out[[1]]]||twinPrimesQ[out[[2]]];
        If[found,Break[]];
    ];
    If[found,out~Select~(twinPrimesQ[#]&)//First,False]
];

This defines a function twinPrime[p,pl] where p=$\,p$ and pl is the maximum power of prime factors of $P_p$ to search upon. The function returns the first found twin pair or False if it has failed.

For example:

You can try this online with Mathics.

To confirm or disprove your conjecture for ranges of primes, you can use

out=List[]; For[i=4,i<=7,i=i+1,out=out~Append~{i,Prime[i],testPrime[Prime[i],1]}]; out//TableForm

adjusting the bounds of search (values of j in terms of consecutive number of primes) and maximum power of prime factors. This will output a table with three columns: the id of the prime being tested, the prime itself and the first twin prime pair found (or False if none).